A pyramid has a base in the shape of a rhombus and a peak directly above the base's center. The pyramid's height is #7 #, its base's sides have lengths of #5 #, and its base has a corner with an angle of #(5 pi)/8 #. What is the pyramid's surface area?

Answer 1

T S A = 97.4271

AB = BC = CD = DA = a = 5
Height OE = h = 7
OF = a/2 = 1/2 = 2.5
# EF = sqrt(EO^2+ OF^2) = sqrt (h^2 + (a/2)^2) = sqrt(7^2+2.5^2) = color(red)(7.433)#

Area of #DCE = (1/2)*a*EF = (1/2)*5*7.433 = color(red)(18.5825)#
Lateral surface area #= 4*Delta DCE = 4*18.5825 = color(blue)(74.33)#

#/_C = (5pi)/8, /_C/2 = (5pi)/16#
diagonal #AC = d_1# & diagonal #BD = d_2#
#OB = d_2/2 = BCsin (C/2)=5sin((5pi)/16)= 4.1573

#OC = d_1/2 = BC cos (C/2) = 5* cos ((5pi)/16) = 2.7779

Area of base ABCD #= (1/2)*d_1*d_2 = (1/2)(2*4.1573) (2*2.7779) = color (blue)(23.0971)#

T S A #= Lateral surface area + Base area#
T S A # =74.33 + 23.0971 = color(purple)(97.4271)#

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Answer 2

To find the surface area of the pyramid, we need to calculate the area of the rhombus base and the area of the four triangular faces.

  1. Area of the rhombus base: The area of a rhombus can be calculated using the formula: ( \text{Area} = \frac{d_1 \times d_2}{2} ), where ( d_1 ) and ( d_2 ) are the diagonals of the rhombus. Given that the sides of the base rhombus have a length of 5, and the angle between two adjacent sides is ( \frac{5\pi}{8} ), we can use trigonometry to find the diagonals. Let ( \theta = \frac{5\pi}{8} ). The diagonals of a rhombus are perpendicular bisectors of each other, so we can use ( \theta ) to find the lengths of the diagonals. ( \text{Diagonal} = 2 \times \text{Side} \times \sin\left(\frac{\theta}{2}\right) ) ( d_1 = 2 \times 5 \times \sin\left(\frac{5\pi}{16}\right) ) ( d_2 = 2 \times 5 \times \sin\left(\frac{3\pi}{16}\right) ) Once we have the lengths of the diagonals, we can find the area of the rhombus using the formula.

  2. Area of each triangular face: The area of a triangle can be calculated using the formula: ( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ). The height of the pyramid can be found using the Pythagorean theorem since it forms a right triangle with the half of one diagonal of the rhombus and the height of the pyramid. Once we have the height of the triangular face, we can use the formula to find the area of each triangular face and multiply it by 4.

  3. Add the area of the rhombus base and the total area of the four triangular faces to find the surface area of the pyramid.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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